On the Non-homogeneity of Completely Positive Cones

For a closed cone C in Rn, the completely positive cone of C is the convex cone K in Sn generated by {uuT : u ∈ C}. Completely positive cones arise, for example, in the conic LP reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints. Motivated by the useful and desirable properties of the nonnegative orthant and the positive semidefinite cone (and more generally of symmetric cones in Euclidean Jordan algebras), in this paper, we investigate when (or whether) K can be self-dual, irreducible, or homogeneous.